Why does Sage say my Groebner basis is 1?
Consider the following program
A.<a1, a2, a3, l2, l3> = PolynomialRing(QQ)
F = A.fraction_field()
F.inject_variables()
R.<c2, s2, c1, s1> = PolynomialRing(F, order = 'lex')
I = R.ideal(l3*s1*s2 + l2*c1 - a1, l3*s1*c2 - l2*s1 - a2, l3*c2 - a3, c1^2 + s1^2 - 1, c2^2 + s2^2 - 1)
I.groebner_basis()
So a1,a2,a3,l2,l3 are parameters, and I am trying to solve the system of equations
a1=l3s1s2+l2c1
a2=l3s1c2−l2s1
a3=l3c2
c21+s11=1
c22+s22=1
Now this gives as result 1, meaning that there is no solution to this system. However, for a1=l2, a2=0 and a3=l3 we do have a solution, namely c1=1,s1=0,c2=1,s2=0.
So is something in the programming going wrong, or am I misunderstanding something on the algebraic geometry part?